where the last step holds by using \({\chi _{\left[ {\kappa } \right]}}\left( {a{\bf {X}}} \right) = {a^k}{\chi _{\left[ {\kappa } \right]}}\left( {\bf {X}} \right)\) . By using [1]}, \({\chi _{\left[ {{{\kappa } _k}} \right]}}\left( {\bf {V}} \right)\)  can be simplified as
\({\chi _{\left[ {{{\kappa } _k}} \right]}}\left( {\bf {V}} \right) &= {\chi _{\left[ {{\kappa }_k^{\prime }} \right]}}\left( {{{\bf {V}}_{\left( {1:{N_r} - M + 1} \right) \times \left( {1:{N_r} - M + 1} \right)}}} \right) \\&= \frac{{{C_{{{\kappa } _k^{\prime }}}}\left( {\bf {I}} \right)}}{{{\chi _{\left[ {{{\kappa } _k^{\prime }}} \right]}}\left( 1 \right)}}=\frac{\left( {{{N_r} - M+1}} \right)_k}{k!}.\) 
